7.5.6 The Homotopy Colimit as a Derived Functor
Let $\operatorname{\mathcal{C}}$ be a small category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be a diagram of $\infty $-categories. In ยง7.5.3, we showed that the homotopy limit $ \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {F} )$ can be identified with the limit of an isofibrant replacement for $\mathscr {F}$: that is, there exists an isomorphism $ \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {F} ) \simeq \varprojlim ( \mathscr {F}^{+} )$, where $\mathscr {F}^{+}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ is an isofibrant diagram equipped with a levelwise categorical equivalence $\mathscr {F} \hookrightarrow \mathscr {F}^{+}$ (Construction 7.5.3.3 and Proposition 7.5.3.7). Our goal in this section is to present a parallel treatment of the homotopy colimit functor of Construction 5.3.2.1. More precisely, we show that the homotopy colimit of a diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ can be identified with the colimit of an auxiliary diagram $\mathscr {F}_{+}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ which is equipped with a levelwise weak homotopy equivalence $\mathscr {F}_{+} \twoheadrightarrow \mathscr {F}$ (Proposition 7.5.6.12).
We begin by introducing some terminology. Recall that a natural transformation $\beta : \widetilde{\mathscr {G}} \rightarrow \mathscr {G}$ is a levelwise trivial Kan fibration if, for each object $C \in \operatorname{\mathcal{C}}$, the morphism $\beta _{C}: \widetilde{\mathscr {G}}(C) \rightarrow \mathscr {G}(C)$ is a trivial Kan fibration of simplicial sets.
Definition 7.5.6.1. Let $\operatorname{\mathcal{C}}$ be a small category. We say that a diagram of simplicial sets $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ is projectively cofibrant if, for every levelwise trivial Kan fibration $\beta : \mathscr {G}' \rightarrow \mathscr {G}$, the induced map
\[ \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}},\operatorname{Set_{\Delta }}) }( \mathscr {F}, \mathscr {G}' ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}},\operatorname{Set_{\Delta }}) }( \mathscr {F}, \mathscr {G}) \]
is surjective. That is, every natural transformation $\alpha : \mathscr {F} \rightarrow \mathscr {G}$ factors through $\beta $.
Example 7.5.6.2. Let $\operatorname{\mathcal{C}}$ be a category and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ be a morphism of simplicial sets. Then the diagram
\[ \mathscr {F}_{\operatorname{\mathcal{E}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}\quad \quad \mathscr {F}_{\operatorname{\mathcal{E}}}(C) = \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{/C} ) \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{\mathcal{E}} \]
is projectively cofibrant, in the sense of Definition 7.5.6.1. To prove this, we must show that for every levelwise trivial Kan fibration $\mathscr {G}' \rightarrow \mathscr {G}$ between functors $\mathscr {G}', \mathscr {G}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$, the induced map
\[ \theta : \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}) }( \mathscr {F}_{\operatorname{\mathcal{E}}}, \mathscr {G}') \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}) }( \mathscr {G}_{\operatorname{\mathcal{E}}}, \mathscr {G} ) \]
is surjective. Using Proposition 5.3.3.24, we can identify $\theta $ with a pullback of the map $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\mathcal{E}}, \operatorname{N}_{\bullet }^{\mathscr {G}'}(\operatorname{\mathcal{C}}) ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\mathcal{E}}, \operatorname{N}_{\bullet }^{\mathscr {G} }(\operatorname{\mathcal{C}}) )$, which is surjective by virtue of Exercise 5.3.3.11.
Exercise 7.5.6.3 (Well-Founded Diagrams). Let $(Q, \leq )$ be a well-founded partially ordered set. Show that a diagram of simplicial sets $\mathscr {F}: Q \rightarrow \operatorname{Set_{\Delta }}$ is projectively cofibrant if and only if, for each element $q \in Q$, the associated map $\varinjlim _{ p < q} \mathscr {F}(p) \rightarrow \mathscr {F}(q)$ is a monomorphism of simplicial sets (compare with Proposition 4.5.6.6).
Example 7.5.6.4 (Projectively Cofibrant Sequences). A sequential diagram of simplicial sets
\[ X(0) \rightarrow X(1) \rightarrow X(2) \rightarrow X(3) \rightarrow \cdots \]
is projectively cofibrant (when regarded as a functor $\operatorname{\mathbf{Z}}_{\geq 0} \rightarrow \operatorname{Set_{\Delta }})$ if and only if each of the transition maps $X(n) \rightarrow X(n+1)$ is a monomorphism.
Example 7.5.6.5 (Projectively Cofibrant Squares). A commutative diagram of simplicial sets
7.53
\begin{equation} \begin{gathered}\label{equation:projectively-cofibrant-squares} \xymatrix@C =50pt@R=50pt{ A \ar [r]^-{f_0} \ar [d]^{f_1} & A_0 \ar [d]^{f_1} \\ A_1 \ar [r]^-{f_0} & A_{01} } \end{gathered} \end{equation}
is projectively cofibrant (when regarded as a functor $[1] \times [1] \rightarrow \operatorname{Set_{\Delta }}$) if and only if the morphisms
\[ f_0: A \rightarrow A_0 \quad \quad f_1: A \rightarrow A_1 \quad \quad (f'_1, f'_0): A_0 {\coprod }_{A} A_1 \rightarrow A_{01} \]
are monomorphisms of simplicial sets. Equivalently, (7.53) is projectively cofibrant if it is a pullback square consisting of monomorphisms.
Proposition 7.5.6.7. Let $\operatorname{\mathcal{C}}$ be a small category and let $\alpha : \mathscr {F} \rightarrow \mathscr {G}$ be a natural transformation between projectively cofibrant diagrams $\mathscr {F}, \mathscr {G}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$. If $\alpha $ is a levelwise categorical equivalence, then the induced map $\varinjlim (\alpha ): \varinjlim ( \mathscr {F} ) \rightarrow \varinjlim ( \mathscr {G} )$ is a categorical equivalence of simplicial sets. If $\alpha $ is a levelwise weak homotopy equivalence, then $\varinjlim (\alpha )$ is a weak homotopy equivalence.
Proof.
We will prove the first assertion; the second follows by a similar argument. Assume that $\alpha $ is levelwise categorical equivalence and let $\operatorname{\mathcal{D}}$ be an $\infty $-category; we wish to show that precomposition with $\varprojlim (\alpha )$ induces an equivalence of $\infty $-categories $\alpha ^{\ast }: \operatorname{Fun}( \varinjlim ( \mathscr {G} ), \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \varinjlim ( \mathscr {F}), \operatorname{\mathcal{D}})$. $\alpha $ is a levelwise categorical equivalence, precomposition with $\alpha $ induces a levelwise categorical equivalence $\beta : \operatorname{\mathcal{D}}^{ \mathscr {G} } \rightarrow \operatorname{\mathcal{D}}^{\mathscr {F}}$ in the category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{Set_{\Delta }})$. Unwinding the definitions, we see that $\alpha ^{\ast }$ can be identified with the limit $\varprojlim (\beta )$. Since $\operatorname{\mathcal{D}}^{\mathscr {F}}$ and $\operatorname{\mathcal{D}}^{\mathscr {G}}$ are isofibrant diagrams (Remark 7.5.6.6), the functor $\varprojlim (\beta )$ is an equivalence of $\infty $-categories (Corollary 4.5.6.17).
$\square$
We now show that every diagram of simplicial sets $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ admits a weak homotopy equivalence from a projectively cofibrant diagram (for a stronger statement, see Proposition 7.5.9.7).
Construction 7.5.6.8 (Explicit Cofibrant Replacement). Let $\operatorname{\mathcal{C}}$ be a small category, let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets, and let $ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )$ denote the homotopy colimit of $\mathscr {F}$ (Construction 5.3.2.1). For each object $C \in \operatorname{\mathcal{C}}$, we let $\mathscr {F}_{+}(C)$ denote the simplicial set given by the fiber product
\[ \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{ /C} ) \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) = \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F}|_{ \operatorname{\mathcal{C}}_{/C} } ). \]
The construction $C \mapsto \mathscr {F}_{+}(C)$ determines a diagram of simplicial sets $\mathscr {F}_{+}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$. This diagram is equipped with a natural transformation $\alpha : \mathscr {F}_{+} \twoheadrightarrow \mathscr {F}$, which carries each object $C \in \operatorname{\mathcal{C}}$ to the comparison map
\[ \mathscr {F}_{+}(C) = \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F}|_{ \operatorname{\mathcal{C}}_{/C} } ) \twoheadrightarrow \varinjlim ( \mathscr {F}|_{ \operatorname{\mathcal{C}}_{/C} } ) \simeq \mathscr {F}(C) \]
of Remark 5.3.2.9.
Proposition 7.5.6.9. Let $\operatorname{\mathcal{C}}$ be a small category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets. Then the diagram $\mathscr {F}_{+}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ of Construction 7.5.6.8 is projectively cofibrant, and the natural transformation $\alpha : \mathscr {F}_{+} \rightarrow \mathscr {F}$ is a levelwise weak homotopy equivalence. Moreover, $\alpha $ is also an epimorphism.
Proof.
Example 7.5.6.2 shows that the diagram $\mathscr {F}_{+}$ is projectively cofibrant and Remark 5.3.2.9 shows that $\alpha $ is an epimorphism. To complete the proof, it will suffice to show that for each object $C \in \operatorname{\mathcal{C}}$, the map $\alpha _{C}: \mathscr {F}_{+}(C) \rightarrow \mathscr {F}(C)$ is a weak homotopy equivalence of simplicial sets. Replacing $\operatorname{\mathcal{C}}$ by the slice category $\operatorname{\mathcal{C}}_{/C}$, we can reduce to the case where $C$ is a final object of $\operatorname{\mathcal{C}}$; in this case, we wish to prove that the comparison map
\[ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \varinjlim ( \mathscr {F} ) \simeq \mathscr {F}(C) \]
is a weak homotopy equivalence. Note that this map admits a section, given by the inclusion map
\[ \iota : \mathscr {F}(C) \simeq \{ C\} \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ). \]
We complete the proof by that our assumption that $C \in \operatorname{\mathcal{C}}$ is a final object guarantees that $\iota $ is right anodyne (Example 7.2.3.11).
$\square$
Warning 7.5.6.10. In the situation of Proposition 7.5.6.9, the natural transformation $\alpha : \mathscr {F}_{+} \twoheadrightarrow \mathscr {F}$ is usually not a levelwise categorical equivalence. For example, if $\mathscr {F}$ is the constant functor taking the value $\Delta ^0$, then $\mathscr {F}_{+}$ is given by the construction $C \mapsto \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{/C} )$.
Proposition 7.5.6.12. Let $\operatorname{\mathcal{C}}$ be a small category, let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets, and let $\mathscr {F}_{+}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be the diagram of Construction 7.5.6.8. Then there is a canonical isomorphism of simplicial sets $\lambda : \varinjlim ( \mathscr {F}_{+} ) \rightarrow \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F})$ which is characterized by the following requirement: for each object $C \in \operatorname{\mathcal{C}}$, the composition
\begin{eqnarray*} \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{/C} ) \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) & = & \mathscr {F}_{+}(C) \\ & \rightarrow & \varinjlim ( \mathscr {F}_{+} ) \\ & \xrightarrow {\lambda } & \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \end{eqnarray*}
is given by projection onto the second factor.
Proof.
It follows from the definition of the colimit that there is a unique morphism of simplicial sets $\lambda : \varinjlim ( \mathscr {F}_{+} ) \rightarrow \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )$ having the desired property. Using the dual of Lemma 7.5.3.8, we deduce that $\lambda $ is an isomorphism.
$\square$
Corollary 7.5.6.14. Let $\operatorname{\mathcal{C}}$ be a small category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a projectively cofibrant diagram of simplicial sets. Then the comparison map $ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \twoheadrightarrow \varinjlim ( \mathscr {F} )$ of Remark 5.3.2.9 is a weak homotopy equivalence.
Proof.
By virtue of Remark 7.5.6.13, it will suffice to show that the natural transformation $\alpha : \mathscr {F}_{+} \twoheadrightarrow \mathscr {F}$ of Construction 7.5.6.8 induces a weak homotopy equivalence $\varinjlim (\alpha ): \varinjlim ( \mathscr {F}_{+} ) \rightarrow \varinjlim ( \mathscr {F} )$. This is a special case of Proposition 7.5.6.7, since $\alpha $ is a levelwise weak homotopy equivalence between projectively cofibrant diagrams (Proposition 7.5.6.9).
$\square$
Warning 7.5.6.15. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be a diagram of simplicial sets, let $\alpha : \mathscr {F}_{+} \twoheadrightarrow \mathscr {F}$ be the natural transformation of Construction 7.5.6.8, and let $\lambda : \varinjlim ( \mathscr {F}_{+} ) \xrightarrow {\sim } \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )$ be the isomorphism of Proposition 7.5.6.12. Then we have a diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F}_{+} ) \ar [r]^-{ \underset { \longrightarrow }{\mathrm{holim}}( \alpha ) } \ar [d] & \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \ar [d] \\ \varinjlim ( \mathscr {F}_{+} ) \ar [r]_{\varinjlim (\alpha ) } \ar [ur]^{\lambda }_{\sim } & \varinjlim ( \mathscr {F} ), } \]
where the outer square and the lower right triangle are commutative (Remark 7.5.6.13). Beware that the upper left triangle is usually not commutative. That is, $ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )$ and $\varinjlim ( \mathscr {F}_{+} )$ are isomorphic when viewed as abstract simplicial sets, but not when viewed as quotients of the simplicial set $ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F}_{+} )$ (compare with Warning 7.5.3.14).
